A very quick way to determine if you'll have a dependent system is noticing that there are more variables than equations.

anonymous

6 years ago

so the answer is consistent and dependent or x = 79z - 30
sorry I am a little confused haha

jim_thompson5910

6 years ago

Well both aspects make up the total answer.
The idea here is that there are an infinite number of points where the two equations x-2y+5z=2 and 4x -z=0 intersect
So this makes the system dependent (since one equation technically "depends" on the other -- it changes as the other one does)
Because at least one solution exists, this makes the system consistent.
The answers of x = 79s-30, etc... are the more explicit forms of the solutions and are more specific than saying "there are an infinite number of solutions"
This is because you can't just say that something random like (1,2,3) is a solution even though I did say "there are an infinite number of solutions". All of the solutions fit a very specific algebraic form.